Chapter 7.5, Problem 9GP

Solution

To state: The recursive rule for the sequence $1,2,2,4,8,32,\cdots$ .

The resultant recursive rule is $a_n=a_{n-2}\cdot a_{n-1}$ .

Given information:

The given sequence is $1,2,2,4,8,32,\cdots$ .

Explanation:

Consider the given sequence,

  $1,2,2,4,8,32,\cdots$

The first term of the sequence is: $a_1=1$

The common difference ( d ) is:

  $\begin{array}{c}2-1=1\\ 2-2=0\\ 4-2=2\end{array}$

The given sequence is not an arithmetic sequence because the common difference of two successive terms is not equal.

Now check if the sequence is geometric.

The common ratio ( r ) is:

  $\begin{array}{c}r=\frac{2}{1},r=2\\ r=\frac{2}{2},r=1\\ r=\frac{4}{2},r=2\end{array}$

The given sequence is not a geometric sequence because the common ratio is not same from the two terms.

Now again consider the sequence $1,2,2,4,8,32,\cdots$ ,

Beginning with the third term in the sequence, $a_3$ can be obtained by multiplying the first term and second term.

Similarly, the fourth term can be obtained by multiplying the second term and third term.

So the recursive rule for the sequence $a_1=1,a_2=2$ is $a_n=a_{n-2}\cdot a_{n-1}$ .